metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.49D28, C4○D28⋊6C4, D28⋊18(C2×C4), C28.419(C2×D4), (C2×C4).154D28, (C2×C8).190D14, (C2×C28).175D4, C2.D56⋊40C2, C4.39(D14⋊C4), Dic14⋊17(C2×C4), C2.5(C8⋊D14), (C2×M4(2))⋊13D7, C22.58(C2×D28), C28.44D4⋊40C2, C14.21(C8⋊C22), C28.28(C22⋊C4), (C14×M4(2))⋊21C2, (C2×C56).320C22, C28.116(C22×C4), (C2×C28).774C23, C22.3(D14⋊C4), C2.5(C8.D14), (C22×C14).102D4, (C22×C4).141D14, C7⋊4(C23.36D4), (C2×D28).201C22, C14.21(C8.C22), C4⋊Dic7.285C22, (C22×C28).190C22, (C2×Dic14).221C22, C4.74(C2×C4×D7), (C2×C4).54(C4×D7), (C2×C4⋊Dic7)⋊33C2, C2.32(C2×D14⋊C4), C4.112(C2×C7⋊D4), (C2×C28).110(C2×C4), (C2×C4○D28).13C2, (C2×C14).164(C2×D4), (C2×C4).78(C7⋊D4), C14.60(C2×C22⋊C4), (C2×C4).723(C22×D7), (C2×C14).22(C22⋊C4), SmallGroup(448,667)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.49D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 868 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C23.36D4, C4⋊Dic7, C4⋊Dic7, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, C28.44D4, C2.D56, C2×C4⋊Dic7, C14×M4(2), C2×C4○D28, C23.49D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8⋊C22, C8.C22, C4×D7, D28, C7⋊D4, C22×D7, C23.36D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C8⋊D14, C8.D14, C2×D14⋊C4, C23.49D28
►On 224 points
Generators in S224
(1 201)(2 174)(3 203)(4 176)(5 205)(6 178)(7 207)(8 180)(9 209)(10 182)(11 211)(12 184)(13 213)(14 186)(15 215)(16 188)(17 217)(18 190)(19 219)(20 192)(21 221)(22 194)(23 223)(24 196)(25 169)(26 198)(27 171)(28 200)(29 173)(30 202)(31 175)(32 204)(33 177)(34 206)(35 179)(36 208)(37 181)(38 210)(39 183)(40 212)(41 185)(42 214)(43 187)(44 216)(45 189)(46 218)(47 191)(48 220)(49 193)(50 222)(51 195)(52 224)(53 197)(54 170)(55 199)(56 172)(57 142)(58 115)(59 144)(60 117)(61 146)(62 119)(63 148)(64 121)(65 150)(66 123)(67 152)(68 125)(69 154)(70 127)(71 156)(72 129)(73 158)(74 131)(75 160)(76 133)(77 162)(78 135)(79 164)(80 137)(81 166)(82 139)(83 168)(84 141)(85 114)(86 143)(87 116)(88 145)(89 118)(90 147)(91 120)(92 149)(93 122)(94 151)(95 124)(96 153)(97 126)(98 155)(99 128)(100 157)(101 130)(102 159)(103 132)(104 161)(105 134)(106 163)(107 136)(108 165)(109 138)(110 167)(111 140)(112 113)
(1 201)(2 202)(3 203)(4 204)(5 205)(6 206)(7 207)(8 208)(9 209)(10 210)(11 211)(12 212)(13 213)(14 214)(15 215)(16 216)(17 217)(18 218)(19 219)(20 220)(21 221)(22 222)(23 223)(24 224)(25 169)(26 170)(27 171)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 141)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 77 173 134)(2 133 174 76)(3 75 175 132)(4 131 176 74)(5 73 177 130)(6 129 178 72)(7 71 179 128)(8 127 180 70)(9 69 181 126)(10 125 182 68)(11 67 183 124)(12 123 184 66)(13 65 185 122)(14 121 186 64)(15 63 187 120)(16 119 188 62)(17 61 189 118)(18 117 190 60)(19 59 191 116)(20 115 192 58)(21 57 193 114)(22 113 194 112)(23 111 195 168)(24 167 196 110)(25 109 197 166)(26 165 198 108)(27 107 199 164)(28 163 200 106)(29 105 201 162)(30 161 202 104)(31 103 203 160)(32 159 204 102)(33 101 205 158)(34 157 206 100)(35 99 207 156)(36 155 208 98)(37 97 209 154)(38 153 210 96)(39 95 211 152)(40 151 212 94)(41 93 213 150)(42 149 214 92)(43 91 215 148)(44 147 216 90)(45 89 217 146)(46 145 218 88)(47 87 219 144)(48 143 220 86)(49 85 221 142)(50 141 222 84)(51 83 223 140)(52 139 224 82)(53 81 169 138)(54 137 170 80)(55 79 171 136)(56 135 172 78)
G:=sub<Sym(224)| (1,201)(2,174)(3,203)(4,176)(5,205)(6,178)(7,207)(8,180)(9,209)(10,182)(11,211)(12,184)(13,213)(14,186)(15,215)(16,188)(17,217)(18,190)(19,219)(20,192)(21,221)(22,194)(23,223)(24,196)(25,169)(26,198)(27,171)(28,200)(29,173)(30,202)(31,175)(32,204)(33,177)(34,206)(35,179)(36,208)(37,181)(38,210)(39,183)(40,212)(41,185)(42,214)(43,187)(44,216)(45,189)(46,218)(47,191)(48,220)(49,193)(50,222)(51,195)(52,224)(53,197)(54,170)(55,199)(56,172)(57,142)(58,115)(59,144)(60,117)(61,146)(62,119)(63,148)(64,121)(65,150)(66,123)(67,152)(68,125)(69,154)(70,127)(71,156)(72,129)(73,158)(74,131)(75,160)(76,133)(77,162)(78,135)(79,164)(80,137)(81,166)(82,139)(83,168)(84,141)(85,114)(86,143)(87,116)(88,145)(89,118)(90,147)(91,120)(92,149)(93,122)(94,151)(95,124)(96,153)(97,126)(98,155)(99,128)(100,157)(101,130)(102,159)(103,132)(104,161)(105,134)(106,163)(107,136)(108,165)(109,138)(110,167)(111,140)(112,113), (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,211)(12,212)(13,213)(14,214)(15,215)(16,216)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,141), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,77,173,134)(2,133,174,76)(3,75,175,132)(4,131,176,74)(5,73,177,130)(6,129,178,72)(7,71,179,128)(8,127,180,70)(9,69,181,126)(10,125,182,68)(11,67,183,124)(12,123,184,66)(13,65,185,122)(14,121,186,64)(15,63,187,120)(16,119,188,62)(17,61,189,118)(18,117,190,60)(19,59,191,116)(20,115,192,58)(21,57,193,114)(22,113,194,112)(23,111,195,168)(24,167,196,110)(25,109,197,166)(26,165,198,108)(27,107,199,164)(28,163,200,106)(29,105,201,162)(30,161,202,104)(31,103,203,160)(32,159,204,102)(33,101,205,158)(34,157,206,100)(35,99,207,156)(36,155,208,98)(37,97,209,154)(38,153,210,96)(39,95,211,152)(40,151,212,94)(41,93,213,150)(42,149,214,92)(43,91,215,148)(44,147,216,90)(45,89,217,146)(46,145,218,88)(47,87,219,144)(48,143,220,86)(49,85,221,142)(50,141,222,84)(51,83,223,140)(52,139,224,82)(53,81,169,138)(54,137,170,80)(55,79,171,136)(56,135,172,78)>;
G:=Group( (1,201)(2,174)(3,203)(4,176)(5,205)(6,178)(7,207)(8,180)(9,209)(10,182)(11,211)(12,184)(13,213)(14,186)(15,215)(16,188)(17,217)(18,190)(19,219)(20,192)(21,221)(22,194)(23,223)(24,196)(25,169)(26,198)(27,171)(28,200)(29,173)(30,202)(31,175)(32,204)(33,177)(34,206)(35,179)(36,208)(37,181)(38,210)(39,183)(40,212)(41,185)(42,214)(43,187)(44,216)(45,189)(46,218)(47,191)(48,220)(49,193)(50,222)(51,195)(52,224)(53,197)(54,170)(55,199)(56,172)(57,142)(58,115)(59,144)(60,117)(61,146)(62,119)(63,148)(64,121)(65,150)(66,123)(67,152)(68,125)(69,154)(70,127)(71,156)(72,129)(73,158)(74,131)(75,160)(76,133)(77,162)(78,135)(79,164)(80,137)(81,166)(82,139)(83,168)(84,141)(85,114)(86,143)(87,116)(88,145)(89,118)(90,147)(91,120)(92,149)(93,122)(94,151)(95,124)(96,153)(97,126)(98,155)(99,128)(100,157)(101,130)(102,159)(103,132)(104,161)(105,134)(106,163)(107,136)(108,165)(109,138)(110,167)(111,140)(112,113), (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,211)(12,212)(13,213)(14,214)(15,215)(16,216)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,141), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,77,173,134)(2,133,174,76)(3,75,175,132)(4,131,176,74)(5,73,177,130)(6,129,178,72)(7,71,179,128)(8,127,180,70)(9,69,181,126)(10,125,182,68)(11,67,183,124)(12,123,184,66)(13,65,185,122)(14,121,186,64)(15,63,187,120)(16,119,188,62)(17,61,189,118)(18,117,190,60)(19,59,191,116)(20,115,192,58)(21,57,193,114)(22,113,194,112)(23,111,195,168)(24,167,196,110)(25,109,197,166)(26,165,198,108)(27,107,199,164)(28,163,200,106)(29,105,201,162)(30,161,202,104)(31,103,203,160)(32,159,204,102)(33,101,205,158)(34,157,206,100)(35,99,207,156)(36,155,208,98)(37,97,209,154)(38,153,210,96)(39,95,211,152)(40,151,212,94)(41,93,213,150)(42,149,214,92)(43,91,215,148)(44,147,216,90)(45,89,217,146)(46,145,218,88)(47,87,219,144)(48,143,220,86)(49,85,221,142)(50,141,222,84)(51,83,223,140)(52,139,224,82)(53,81,169,138)(54,137,170,80)(55,79,171,136)(56,135,172,78) );
G=PermutationGroup([[(1,201),(2,174),(3,203),(4,176),(5,205),(6,178),(7,207),(8,180),(9,209),(10,182),(11,211),(12,184),(13,213),(14,186),(15,215),(16,188),(17,217),(18,190),(19,219),(20,192),(21,221),(22,194),(23,223),(24,196),(25,169),(26,198),(27,171),(28,200),(29,173),(30,202),(31,175),(32,204),(33,177),(34,206),(35,179),(36,208),(37,181),(38,210),(39,183),(40,212),(41,185),(42,214),(43,187),(44,216),(45,189),(46,218),(47,191),(48,220),(49,193),(50,222),(51,195),(52,224),(53,197),(54,170),(55,199),(56,172),(57,142),(58,115),(59,144),(60,117),(61,146),(62,119),(63,148),(64,121),(65,150),(66,123),(67,152),(68,125),(69,154),(70,127),(71,156),(72,129),(73,158),(74,131),(75,160),(76,133),(77,162),(78,135),(79,164),(80,137),(81,166),(82,139),(83,168),(84,141),(85,114),(86,143),(87,116),(88,145),(89,118),(90,147),(91,120),(92,149),(93,122),(94,151),(95,124),(96,153),(97,126),(98,155),(99,128),(100,157),(101,130),(102,159),(103,132),(104,161),(105,134),(106,163),(107,136),(108,165),(109,138),(110,167),(111,140),(112,113)], [(1,201),(2,202),(3,203),(4,204),(5,205),(6,206),(7,207),(8,208),(9,209),(10,210),(11,211),(12,212),(13,213),(14,214),(15,215),(16,216),(17,217),(18,218),(19,219),(20,220),(21,221),(22,222),(23,223),(24,224),(25,169),(26,170),(27,171),(28,172),(29,173),(30,174),(31,175),(32,176),(33,177),(34,178),(35,179),(36,180),(37,181),(38,182),(39,183),(40,184),(41,185),(42,186),(43,187),(44,188),(45,189),(46,190),(47,191),(48,192),(49,193),(50,194),(51,195),(52,196),(53,197),(54,198),(55,199),(56,200),(57,142),(58,143),(59,144),(60,145),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125),(97,126),(98,127),(99,128),(100,129),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,141)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,77,173,134),(2,133,174,76),(3,75,175,132),(4,131,176,74),(5,73,177,130),(6,129,178,72),(7,71,179,128),(8,127,180,70),(9,69,181,126),(10,125,182,68),(11,67,183,124),(12,123,184,66),(13,65,185,122),(14,121,186,64),(15,63,187,120),(16,119,188,62),(17,61,189,118),(18,117,190,60),(19,59,191,116),(20,115,192,58),(21,57,193,114),(22,113,194,112),(23,111,195,168),(24,167,196,110),(25,109,197,166),(26,165,198,108),(27,107,199,164),(28,163,200,106),(29,105,201,162),(30,161,202,104),(31,103,203,160),(32,159,204,102),(33,101,205,158),(34,157,206,100),(35,99,207,156),(36,155,208,98),(37,97,209,154),(38,153,210,96),(39,95,211,152),(40,151,212,94),(41,93,213,150),(42,149,214,92),(43,91,215,148),(44,147,216,90),(45,89,217,146),(46,145,218,88),(47,87,219,144),(48,143,220,86),(49,85,221,142),(50,141,222,84),(51,83,223,140),(52,139,224,82),(53,81,169,138),(54,137,170,80),(55,79,171,136),(56,135,172,78)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D28 | C8⋊C22 | C8.C22 | C8⋊D14 | C8.D14 |
| kernel | C23.49D28 | C28.44D4 | C2.D56 | C2×C4⋊Dic7 | C14×M4(2) | C2×C4○D28 | C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 6 | 3 | 12 | 6 | 12 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C23.49D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 18 | 0 | 1 | 0 |
| 0 | 0 | 18 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 73 | 28 | 0 | 0 | 0 | 0 |
| 57 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 112 | 15 | 0 |
| 0 | 0 | 44 | 0 | 15 | 0 |
| 86 | 5 | 0 | 0 | 0 | 0 |
| 80 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 66 | 0 | 0 |
| 0 | 0 | 58 | 44 | 0 | 0 |
| 0 | 0 | 80 | 84 | 17 | 86 |
| 0 | 0 | 0 | 84 | 86 | 96 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,18,18,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[73,57,0,0,0,0,28,59,0,0,0,0,0,0,98,0,0,44,0,0,0,0,112,0,0,0,36,112,15,15,0,0,0,1,0,0],[86,80,0,0,0,0,5,27,0,0,0,0,0,0,69,58,80,0,0,0,66,44,84,84,0,0,0,0,17,86,0,0,0,0,86,96] >;
C23.49D28 in GAP, Magma, Sage, TeX
C_2^3._{49}D_{28} % in TeX
G:=Group("C2^3.49D28"); // GroupNames label
G:=SmallGroup(448,667);
// by ID
G=gap.SmallGroup(448,667);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations